Regularity of the free boundary for the two phase Bernoulli problem - - PowerPoint PPT Presentation

Regularity of the free boundary for the two phase Bernoulli problem

• G. De Philippis

(j/w L. Spolaor, B. Velichkov)

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• G. De Philippis (CIMS): Two phase Bernoulli problem

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• G. De Philippis (CIMS): Two phase Bernoulli problem

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• G. De Philippis (CIMS): Two phase Bernoulli problem

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• G. De Philippis (CIMS): Two phase Bernoulli problem

Working with Alessio... ..to nowadays!

• G. De Philippis (CIMS): Two phase Bernoulli problem

The Bernoulli Free Boundary Problem

Let λ0, λ+, λ− ≥ 0 be given and for D ⊂ Rd let us consider J(u, D) =

• D

|∇u|2 + λ+|{u > 0}| + λ−|{u < 0}| + λ0|{u = 0}|. and the minimization problem (TPBP) min

u|∂D=g J(u, D).

where g is a given function.

• G. De Philippis (CIMS): Two phase Bernoulli problem

The Bernoulli Free Boundary Problem: some remarks

A few simple properties.

• Minimizers are easily seen to exist.
• Uniqueness in general fails.
• A minimizers would like to be harmonic where it is = 0, but the

functional might penalize to be always non zero and/or might impose a “balance” between the negative and positive phase

• G. De Philippis (CIMS): Two phase Bernoulli problem

The Bernoulli Free Boundary Problem: some remarks

When λ0, λ− = 0 and g ≥ 0, the problem reduces to the one phase free boundary problem: (OPBP) min

u=g, u≥0

• J(u, D)
• J(u, D) :=
• D

|∇u|2 + λ+|{u > 0}|

• G. De Philippis (CIMS): Two phase Bernoulli problem

Motivations

These problems have been introduced in the 80’s by Alt-Caffarelli (OPBP) and by Alt-Caffarelli-Friedmann (TPBP) motivated by some problems in flows with jets and cavities. Since then they have been the model problems for a huge class of free boundary problems. More recently these types of problems turned out to have applications in the study of shape optimization problems.

• G. De Philippis (CIMS): Two phase Bernoulli problem

Shape Optimization Problems

Let us consider the following minimization problem: min

U⊂D Cap(U, D) − λ|U|

where Cap(U, D) = min

• D

|∇u|2 u ∈ W 1,2 (D), u = 1 on U

• is the Newtonian capacity of U relative to D. The problem is equivalent to

min

v∈W 1,2 (D)

• D

|∇v|2 − λ|{v = 1}| = min

v∈W 1,2 (D)

• D

|∇v|2 + λ|{0 < v < 1}| − λ|D|. u = 1 − v solves a one phase problem.

• G. De Philippis (CIMS): Two phase Bernoulli problem

Shape Optimization Problems

Let us consider the following minimal partition problem: min

• i

λ(Di) + mi|Di| Di ⊂ D, Di ∩ Dj = ∅ if i = j

• .

Here λ(Di) is the first eigenvalue of the Dirichlet Laplacian on Di, i.e. λ(Di) = inf

• Di |∇u|2
• Di u2

: u ∈ W 1,2 (Di)

• .
• G. De Philippis (CIMS): Two phase Bernoulli problem

Shape Optimization Problems

How minimizers look like? One can show (Spolaor-Trey-Velichkov):

• There are no triple points ∂Di ∩ ∂Dj ∩ ∂Dk = ∅.
• If ui, uj are the first (positive) eigenfunctions of Di, Dj then

v = ui − uj is a (local) minimizer of

• |∇v|2 + mi|{v > 0}| + mj|{v < 0}| + H.O.T.
• G. De Philippis (CIMS): Two phase Bernoulli problem

Back to the Bernoulli free boundary problem

We are interested in the regularity of u and of the free boundary: Γ = Γ+ ∪ Γ− Γ+ = ∂{u > 0} Γ− = ∂{u < 0}. u > 0 u < 0 u = 0 u = 0 Γ+ Γ−

• G. De Philippis (CIMS): Two phase Bernoulli problem

Known results

• u is Lipschitz, Alt-Caffarelli (one phase), Alt-Caffarelli-Friedmann

(two-phase).

• If u is a solution of the one-phase problem, then Γ+ is smooth outside

a (relatively) closed set Σ+ with dimH ≤ d − 5 (Alt-Caffarelli, Weiss, Jerison-Savin, a recent new proof from De Silva).

• There is a minimizer in dimension d = 7 with a point singularity (De

Silva-Jerison).

• If u is a solution of the two phase problem and λ0 ≥ min{λ+, λ−},

then Γ+ = Γ− = Γ is smooth. (Alt-Caffarelli-Friedmann, Caffarelli, De Silva-Ferrari-Salsa).

• G. De Philippis (CIMS): Two phase Bernoulli problem

The case λ0 ≥ min{λ+, λ−}

If λ− ≤ λ0, let v be the harmonic function which is equal to u− on ∂(D \ {u > 0}). Then w = u+ − v satisfies J(w, D) ≤ J(u, D). since λ−|{w < 0}| ≤ λ−|{u < 0}| + λ0|{u = 0}| and

• |∇v|2 ≤
• |∇u−|2

u > 0 u < 0 u > 0 w < 0

• G. De Philippis (CIMS): Two phase Bernoulli problem

The case λ0 < min{λ+, λ−}

When λ0 < min{λ+, λ−} the three phases may co-exist and branch points might appear. u > 0 u < 0 u = 0 u = 0 Branch points P Q

• G. De Philippis (CIMS): Two phase Bernoulli problem

Main result

Theorem D.-Spolaor-Velichkov ’19 (Spolaor-Velichkov’16 for d = 2) Let u be a local minimizer of J. Let us define Γ± = ∂{±u > 0} ΓDP = Γ+ ∩ Γ− Γ±

OP = Γ± \ ΓDP,

Then

• Γ± are C 1,α manifolds outside relatively closed set Σ± with

dimH(Σ±) ≤ d − 5.

• ΓDP ∩ Σ± = ∅. In particular ΓDP is a closed subset of a C 1,α graph.

u > 0 u < 0 ΓDP Γ+

OP

Γ−

OP

u = 0

• G. De Philippis (CIMS): Two phase Bernoulli problem

Steps in the proof

As it is customary in Geometric Measure Theory, the above result is based

• n two steps:
• Blow up analysis.
• ε-regularity theorem.
• G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality conditions

Before detailing the proof, let us start by deriving the optimality conditions for minimizers. The first (trivial) one, one is that u is harmonic where = 0 (which is open) ∆u = 0

• n {u = 0}

What are the optimality conditions on the free boundary? They can be formally obtained by performing inner variations d dε

• ε=0J(uε) = 0

uε(x) = u(x + εX(x)) X ∈ Cc(D; Rd)

• G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition

Let us assume that u is one dimensional:

1 −1 α −β u = αx+ − βx−

• G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition

Let us assume that u is one dimensional:

1 −1 α −β u = αx+ − βx− 1 −1 α −β ε uε = α 1 − ε(x − ε)+ − β 1 + ε(x − ε)−

0 ≤ J(uε) − J(u) = (α2 − β2)ε − (λ+ − λ−)ε + o(ε)

• G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition

Moreover

1 −1 α −β u = αx+ − βx−

• G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition

Moreover

1 −1 α −β u = αx+ − βx− 1 −1 α −β ε uε = α 1 − ε(x − ε)+ − βx−

• G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality condition

Moreover

1 −1 α −β u = αx+ − βx− 1 −1 α −β ε uε = α 1 − ε(x − ε)+ − βx−

0 ≤ J(uε) − J(u) = α2 (1 − ε) − α2 − (λ+ − λ0)ε = α2ε − (λ+ − λ0)ε + o(ε)

• G. De Philippis (CIMS): Two phase Bernoulli problem

Optimality conditions

We get the following problem            ∆u = 0

• n {u = 0}

|∇u±|2 = λ± − λ0

• n Γ±

OP

|∇u+|2 − |∇u−|2 = λ+ − λ−

• n ΓDP

|∇u±|2 ≥ λ± − λ0

• n Γ±
• G. De Philippis (CIMS): Two phase Bernoulli problem

Blow up analysis

The first step consists in understanding which is the asymptotic behavior of the function and of the free boundary.

• G. De Philippis (CIMS): Two phase Bernoulli problem

Blow up analysis

The first step consists in understanding which is the asymptotic behavior of the function and of the free boundary. Let x0 ∈ Γ and r > 0. Let ux0,r(x) = u(x0 + rx) r (u(x0) = 0). Then {ux0,r}r>0 is pre-compact in C 0 and every limit point is

• ne-homogeneous (Weiss Monotonicity Formula).
• G. De Philippis (CIMS): Two phase Bernoulli problem

Blow up analysis

The first step consists in understanding which is the asymptotic behavior of the function and of the free boundary. Let x0 ∈ Γ and r > 0. Let ux0,r(x) = u(x0 + rx) r (u(x0) = 0). Then {ux0,r}r>0 is pre-compact in C 0 and every limit point is

• ne-homogeneous (Weiss Monotonicity Formula).

If x0 ∈ Γ is regular it is easy to see that there is a unique limit vx0 and vx0 =      ±

• λ± − λ0(x · ex0)±

if x0 ∈ Γ±

OP

α+(x · ex0)+ − α−(x · ex0)− if x0 ∈ ΓDP α± ≥

• λ± − λ0,

α2

+ − α2 − = λ+ − λ−

where ex0 is the normal to Γ at x0.

• G. De Philippis (CIMS): Two phase Bernoulli problem

Regular points

We are going to call a point regular if {ux0,r} admits one limit point of the above form (for some e).

1 One can prove that the complement of regular point has small

dimension (Federer dimension reduction) and it does not intersect the two phase free boundary (these are Σ±).

2 The difficult part consists in proving that if x0 is regular the Γ has the

desired structure in a neighborhood, in particular all blow-up coincide. (ε-regularity theory).

3 The ε regularity theory was known at one phase points (Alt-Caffarelli,

De Silva) and at points which are at the interior of the two phase free boundary (Caffarelli, De Silva-Ferrari-Salsa)

4 The new step is to understand what happens at branch points and to

put everything together.

• G. De Philippis (CIMS): Two phase Bernoulli problem

The ε-regularity theorem at one phase point

Let us show De Silva’s proof at one-phase points (λ+ = 1, λ−, λ0 = 0, e = e1). Assume that in B1 u+ ≈ (x1)+ u+ = x1+εvε

• n {u > 0}

ε := u+−x1L∞({u>0}∩B1) What are the equation satisfied by vε?

• G. De Philippis (CIMS): Two phase Bernoulli problem

The ε-regularity theorem at one phase point

Let us show De Silva’s proof at one-phase points (λ+ = 1, λ−, λ0 = 0, e = e1). Assume that in B1 u+ ≈ (x1)+ u+ = x1+εvε

• n {u > 0}

ε := u+−x1L∞({u>0}∩B1) What are the equation satisfied by vε? ∆vε = 0

• n {u > 0} ≈ B+

1 .

moreover 1 = |∇u+|2 = 1 + ε∂1vε + o(ε)

• n ∂{u > 0}

i.e. ∂1vε ≈ 0

• n ∂{u > 0} ≈ {x1 = 0}
• G. De Philippis (CIMS): Two phase Bernoulli problem

The ε-regularity theorem at one phase point

In other words vε is almost a solution of a Neumann problem (NP)

• ∆v = 0
• n B+

1

∂1v = 0

• n {x1 = 0} ∩ B1

The C 2 regularity theory for the (NP) allows to show the existence of Sd−1 ∋ e = e1 + ε∇v(0) + O(ε2)

• e1 ⊥ ∇v(0)
• such that for ρ, δ ≪ 1

u+ − (x · e)+L∞({u>0}∩Bρ) ≤ ρ2−δu+ − (x1)+L∞({u>0}∩B1).

• G. De Philippis (CIMS): Two phase Bernoulli problem

What happens at branch points?

Assume λ± = 1, λ0 = 0. At branch points u ≈ (x1)+ − (x1)− + εv+

ε + εv− ε

• G. De Philippis (CIMS): Two phase Bernoulli problem

What happens at branch points?

Assume λ± = 1, λ0 = 0. At branch points u ≈ (x1)+ − (x1)− + εv+

ε + εv− ε

The functions v±

ε are almost solutions of a thin two membrane problem

(this was first observed by Andersson-Shahgholian-Weiss).         

∆u = 0

• n {u = 0}

|∇u±|2 = 1

• n Γ±

OP

|∇u+|2 = |∇u−|2

• n ΓDP

|∇u±|2 ≥ 1

• n Γ±

⇒         

∆v ± = 0

• n B±

1

∂1v ± = 0

• n {v + = v −} ∩ {x1 = 0}

∂1v + = ∂1v −

• n {v + = v −} ∩ {x1 = 0}

∂1v ± ≥ 0

• n {x1 = 0}

C 1, 1

2 regularity for the two membrane problem would allow to conclude

(same caveat).

• G. De Philippis (CIMS): Two phase Bernoulli problem

The ε-regularity theorem at one phase point: compactness

The key point to make the above proofs rigorous is compactness of v±

ε .

• G. De Philippis (CIMS): Two phase Bernoulli problem

The ε-regularity theorem at one phase point: compactness

The key point to make the above proofs rigorous is compactness of v±

ε .

A good topology is C 0 (solutions will be intended in the viscosity sense) which is the topology where the sequences are bounded. Some a-priori regularity theory is needed (De Silva: adapt Savin’s “Partial Harnack inequality”).

• G. De Philippis (CIMS): Two phase Bernoulli problem

Compactness at branch points

In order to prove compactness one does not only to deal with the case where u ≈ (x1)+ − (x1)− but also u ≈ (x1 + δ1)+ − (x1 + δ1)− with δ1, δ2 ≪ 1. This is the behavior close to branch points. Indeed this is the local picture close a branch point:

u > 0 u < 0 u > 0 u < 0

• G. De Philippis (CIMS): Two phase Bernoulli problem

Congratulations Alessio for this well deserved achievement...

...and let us cheer to the next ones!

Thank you for your attention!

• G. De Philippis (CIMS): Two phase Bernoulli problem